pole

Let $U\subset ℂ$ be a domain and let $a\in ℂ$. A function $f:U\to ℂ$ has a pole at $a$ if it can be represented by a Laurent series centered about $a$ with only finitely many terms of negative exponent; that is,

$$f(z)=\sum _{k=-n}^{\mathrm{\infty }}{c}_k{(z-a)}^k$$

in some nonempty deleted neighborhood of $a$, with $c_{-n}\ne 0$, for some $n\in \mathbb{N}$. The number $n$ is called the order of the pole. A simple pole is a pole of order 1.